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MTE 1: Calculus

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Title Name	IGNOU MTE 1 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 1
Subject Name	Calculus
Year	2026
Session	-
Language	English Medium
Assignment Code	MTE 1/Assignment-1/2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest MTE 01 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Format	Ready-to-Print PDF (.soft copy)

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MTE 1 2025 - English

Assignment

(To be done after studying all the blocks)

Course Code: MTE-01

Assignment Code: MTE-01/TMA/2025

Maximum Marks: 100

1. Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate.

a) The set {S ∈R :x² - 3x + 2 = 0}is an infinite set.

b) The greatest interger function is continuous on R.

c) $frac{d}{dx}left[int_{3}^{e^x}ln t,dt ight]=xe^x-ln 3.$

d) Every integrable function is monotonic.

e) a⊕b = a + b defines a binary operation on Q, the set of rational numbers.

2. a) Find the domain of the function f given by f(x) = $sqrt{frac{2-x}{x^2+1}}$

b) The set R of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on R as:

$a*b=frac{a+b}{2},forall a,binmathbb{R}.$

Is (*) associative in R? Is (.) distributive (*) in R? Check.

3. a) If z-1+2i|= 4, show that the point z+ i describes a circle. Also draw this circle.

b) Express $frac{x-1}{x^3-x^2-2x}$ as a sum of partial fractions.

4. a) Find the least value of a² sec² x + b² cosec²x, where a > 0,b>0.

b) Evaluate:

$intfrac{x^{2}cot^{-1}(x^{3})}{1+x^{6}}dx$

c) For any two sets S and T, show that:

S∪T = (S − T ) ∪ ( S ∩ T) ∪ (T − S).

Depict this situation in the Venn diagram.

5. a) Let f and g be two functions defined on R by:

$f(x)=x^{3}-x^{2}-8x+12$

$g(x)=egin{cases}frac{f(x)}{x+3},& ext{when}x eq-3\alpha,& ext{when}x=-3end{cases}$

i) Find the value of αfor which f is continuous at x = − .3

ii) Find all the roots of f(x) = 0.

b) Find the area between the curve y²(4-x) = x³and its asymptote parallel to y-axis.

6. a) If the revenue function is given by $frac{dR}{dx}=15+2x-x^{2},quad x$ being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0.

b) Trace the curve y² (x+1) = x² (3-x) clearly stating all the properties used for tracing it.

7. a) Find the length of the cycloid x = α(θ −sinθ),y = α 1( − cosθ) and show that the line θ = 2π/3 divides it in the ratio 1 : 3.

b) Find the condition for the curves, ax² by² =1 and a x² + b y ²= 1 intersecting orthogonally.

8. a) If y=e^m sin ^-1X, then show that $(1-x^2)y_{2}-xy_{1}-m^2y=0$ . Hence using Leibnitz’s formula, find the value of $(1-x^2)y_{n+2}-(2n+1)xy_{n+1}$ .

b) Find the largest subset of R on which the function :f R → R defined as:

$f(x)=egin{cases}2x&,x>5\x+5&,1leq xleq 5\|x|&,x<1end{cases}$

is continuous.

9. a) Solve the equation:

$x^{4}+15x^{3}+70x^{2}+120x+64=0$

given that its roots are in G.P.

b) Evaluate:

$lim_{x o 0}frac{ an x-sin x}{x^3}$

10. a) If $I_{m,n}=int x^3(log x)^n,dx$ , Show that:

$(m+1)I_{m,n}=x^{m+1}(log x)^{n}-nI_{m,n-1}$

Hence find the value of $int x^4(log x)^3,dx$ .

b) Verigy Lagrange’s mean value theorem for the function f defined by

$f(x)=2x^2-7x-10quad ext{over}quad[2,5]$

MTE 1 2026 - English

Assignment
(To be done after studying all the blocks)

Course Code: MTE-01
Assignment Code: MTE-01/TMA/2026
Maximum Marks: 100

1. Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate.

a) The set $\{x \in \mathbf{R} : x^2 - 3x + 2 = 0\}$ is an infinite set.

b) The greatest interger function is continuous on $\mathbf{R}$ .

c) $\frac{d}{dx} \left[ \int_{3}^{e^x} \ln t \, dt \right] = x e^x - \ln 3$ .

d) Every integrable function is monotonic.

e) $a \oplus b = \sqrt{a + b}$ defines a binary operation on $\mathbf{Q}$ , the set of rational numbers.

2. a) Find the domain of the function f given by $f(x) = \sqrt{\frac{2-x}{x^2+1}}$ .

b) The set $\mathbf{R}$ of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on $\mathbf{R}$ as:

$$a * b = \frac{a+b}{2}, \quad \forall a, b \in \mathbf{R}.$

Is (*) associative in $\mathbf{R}$ ? Is (.) distributive over (*) in $\mathbf{R}$ ? Check.

3. a) If $|z - 1 + 2i| = 4$ , show that the point z + i describes a circle. Also draw this circle.

b) Express $\frac{x-1}{x^3 - x^2 - 2x}$ as a sum of partial fractions.

4. a) Find the least value of $a^2 \sec^2 x + b^2 \text{cosec}^2 x$ , where a > 0, b > 0.

b) Evaluate:

$$\int \frac{x^2 \cot^{-1}(x^3)}{1 + x^6} dx$

c) For any two sets S and T, show that:

$$S \cup T = (S - T) \cup (S \cap T) \cup (T - S).$

Depict this situation in the Venn diagram.

5. a) Let f and g be two functions defined on R by:

$f(x) = x^3 - x^2 - 8x + 12$

and $g(x) = \begin{cases} \frac{f(x)}{x+3}, & \text{when } x \neq -3 \\ \alpha, & \text{when } x = -3 \end{cases}$

i) Find the value of $\alpha$ for which f is continuous at $x = -3$ .

ii) Find all the roots of $f(x) = 0$ .

b) Find the area between the curve $y^2(4 - x) = x^3$ and its asymptote parallel to y-axis.

6. a) If the revenue function is given by $\frac{dR}{dx} = 15 + 2x - x^2$ , x being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0.

b) Trace the curve $y^2(x + 1) = x^2(3 - x)$ , clearly stating all the properties used for tracing it.

7. a) Find the length of the cycloid $x = \alpha(\theta - \sin \theta), y = \alpha(1 - \cos \theta)$ and show that the line $\theta = \frac{2\pi}{3}$ divides it in the ratio 1 : 3.

b) Find the condition for the curves, $ax^2 + by^2 = 1$ and $a'x^2 + b'y^2 = 1$ intersecting orthogonally.

8. a) If $y = e^{m \sin^{-1} x}$ , then show that $(1 - x^2)y_2 - xy_1 - m^2y = 0$ . Hence using Leibnitz's formula, find the value of (1 - x²)y_n+2 - (2n + 1)xy_n+1.

b) Find the largest subset of **R** on which the function $f : \mathbf{R} \to \mathbf{R}$ defined as:

$$f(x) = \begin{cases} 2x, & x > 5 \\ x + 5, & 1 \le x \le 5 \\ |x|, & x < 1 \end{cases}$

is continuous.

9. a) Solve the equation:

$$x^4 + 15x^3 + 70x^2 + 120x + 64 = 0$

given that its roots are in G.P.

b) Evaluate:

$$\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$

10. a) If $I_{m,n} = \int x^m (\log x)^n dx$ , show that:

$$(m+1)I_{m,n} = x^{m+1} (\log x)^n - n I_{m, n-1}.$
Hence find the value of $\int x^4 (\log x)^3 dx$ .

b) Verify Lagrange's mean value theorem for the function f defined by $f(x) = 2x^2 - 7x - 10$ over [2, 5].

MTE 1 2025 - Hindi

सत्रीय कार्य (सभी ब्लॉकों का अध्ययन करने के बाद किया जाना है)

पाठ्यक्रम कोड: मते-01

सत्रीय कार्य कोड: मते-01/त्मा/2025 अधिकतम अंक: 100

1. निम्नलिखित कथनों में से कौन-से कथन सत्य और कौन-से असत्य हैं? अपने उत्तर के पक्ष में एक संक्षिप्त उपपत्ति या प्रति-उदाहरण दीजिए।

a) समुच्चय {S ∈R :x² - 3x + 2 = 0} एक अपरिमित समुच्चय है।

b) अधिकतम पूर्णांक फलन, आर पर सतत् होता है।

c) $frac{d}{dx}left[int_{3}^{e^x}ln t,dt ight]=xe^x-ln 3.$

d) प्रत्येक समाकलनीय फलन एकदिष्ट होता है।

e) $Aigoplus B=sqrt{a+b}$ परिमेय संख्याओं के समुच्चय क, पर एक द्विआधारी संक्रिया है।

2. a) f(x) = $sqrt{frac{2-x}{x^2+1}}$ द्वारा परिभाषित फलन f का प्रांत ज्ञात कीजिए I

b) वास्तविक संख्याओं का समुच्चय B और उस पर सामान्य जोड़ (+) तथा सामान्य गुणनफल (.) दिए गये हैं। (*), R पर निम्नलिखित से परिभाषित है :

$a*b=frac{a+b}{2},forall a,binmathbb{R}.$

क्या (*), R सहयोगी है? क्या (.), R में (*) पर वितरित है? जाँच कीजिए।

3. a) यदि If z-1+2i|= 4 है, तो दर्शाइए कि बिन्दु z+i एक वृत्त निरूपित करता है। इस वृत्त को खींचिए।

b) $frac{x-1}{x^3-x^2-2x}$ को आंशिक भिन्नों के योग में व्यक्त कीजिए।

4. a) a² sec² x + b² cosec²x, जहाँ a > 0,b>0. हैं, का न्यूनतम मान ज्ञात कीजिए।

b) $intfrac{x^{2}cot^{-1}(x^{3})}{1+x^{6}}dx$ का मान ज्ञात कीजिए।

c) दो समुच्चयों S और T के लिए दर्शाइए किः

S∪T = (S − T ) ∪ ( S ∩ T) ∪ (T − S).

है। वेन आरेख में भी स्थिति दर्शाइए।

5. a) R पर $f(x)=x^{3}-x^{2}-8x+12$ और ज

और $g(x)=egin{cases}frac{f(x)}{x+3},& ext{when}x eq-3\alpha,& ext{when}x=-3end{cases}$

द्वारा परिभाषित दो फलन f और g लीजिए।

i) अ का वह मान ज्ञात कीजिए जिसके लिए फ, ऍक्स = -3 पर सतत् है।

ii) f(x) = 0 के सभी मूल ज्ञात कीजिए।

b) वक्र y²(4-x) = x³ और इसकी y-अक्ष के समांतर अनंतस्पर्शी के बीच का क्षेत्रफल ज्ञात कीजिए।

6. a) यदि एक आय फलन $frac{dR}{dx}=15+2x-x^{2},quad x$ द्वारा दिया गया है, जहाँ x निवेश है, तो

अधिकतम आय ज्ञात कीजिए। यदि प्रारम्भिक आय 0 है, तो आय फलन R भी ज्ञात कीजिए।

b) वक्र y² (x + 1) = x² (3 - x) का आरेखण कीजिए और ऐसा करने के लिए प्रयोग किए गये गुणधर्म भी लिखिए।

7. a) चक्रज x = α(θ −sinθ),y = α 1( − cosθ) की लम्बाई ज्ञात कीजिए और दर्शाइए कि रेखा θ = 2π/3 इसे 1:3 के अनुपात: में विभक्त करती है।

b) वह प्रतिबंध ज्ञात कीजिए कि वक्र ax² by² =1 और a x² + b y ²= 1 एक-दूसरे को लम्बवत् प्रतिच्छेद करते हैं।

8. a) यदि y=e^m sin ^-1X है, तो दर्शाइए कि $(1-x^2)y_{2}-xy_{1}-m^2y=0$ है। इस प्रकार लाइब्नित्ज के सूत्र का प्रयोग करके $(1-x^2)y_{n+2}-(2n+1)xy_{n+1}$ का मान निकालिए।

b) $f(x)=egin{cases}2x&,x>5\x+5&,1leq xleq 5\|x|&,x<1end{cases}$ द्वारा परिभाषित R फलन फ: R आर के जिस भी सबसे बड़े f : R → R समुच्चय पर सतत् है वह निकालिए।

9. a) समीकरण $x^{4}+15x^{3}+70x^{2}+120x+64=0$ हल कीजिए, जिसके सभी मूल ग.पी. में हैं

b) $lim_{x o 0}frac{ an x-sin x}{x^3}$ ज्ञात कीजिए।

10. a) यदि $I_{m,n}=int x^3(log x)^n,dx$ , है, तो दर्शाइए कि :

$f(x)=2x^2-7x-10quad ext{over}quad[2,5]$

MTE 1 2026 - Hindi

सत्रीय कार्य

(सभी ब्लॉकों का अध्ययन करने के बाद किया जाना है)

पाठ्यक्रम कोड: MTE-01

सत्रीय कार्य कोड : MTE-01/TMA/2026

अधिकतम अंक: 100

1. निम्नलिखित कथनों में से कौन-से कथन सत्य और कौन-से असत्य हैं? अपने उत्तर के पक्ष में एक संक्षिप्त उपपत्ति या प्रति-उदाहरण दीजिए।

a) समुच्चय $S = \{x \in \mathbf{R} : x^2 - 3x + 2 = 0\}$ एक अपरिमित समुच्चय है।

b) अधिकतम पूर्णांक फलन, $\mathbf{R}$ पर सतत् होता है।

c) $\frac{d}{dx} \left[ \int_{3}^{e^x} \ln t \, dt \right] = x e^x - \ln 3$ .

d) प्रत्येक समाकलनीय फलन एकदिष्ट होता है।

e) $a \oplus b = \sqrt{a + b}$ परिमेय संख्याओं के समुच्चय $\mathbf{Q}$ , पर एक द्विआधारी संक्रिया है।

2. a) $f(x) = \sqrt{\frac{2 - x}{x^2 + 1}}$ द्वारा परिभाषित फलन f का प्रांत ज्ञात कीजिए।

b) वास्तविक संख्याओं का समुच्चय $\mathbf{R}$ और उस पर सामान्य जोड़ (+) तथा सामान्य गुणनफल $(\cdot)$ दिए गये हैं। $(*), \mathbf{R}$ पर निम्नलिखित से परिभाषित है :
$a * b = \frac{a + b}{2}, \forall a, b \in \mathbf{R}.$
क्या $(*), \mathbf{R}$ सहयोगी है? क्या $(\cdot), \mathbf{R}$ में (*) पर वितरित है? जाँच कीजिए।

3. a) यदि $|z - 1 + 2i| = 4$ है, तो दर्शाइए कि बिन्दु z + i एक वृत्त निरूपित करता है। इस वृत्त को खींचिए।

b) $\frac{x - 1}{x^3 - x^2 - 2x}$ को आंशिक भिन्नों के योग में व्यक्त कीजिए।

4. a) $a^2 \sec^2 x + b^2 \text{cosec}^2 x$ , जहाँ a > 0, b > 0 हैं, का न्यूनतम मान ज्ञात कीजिए।

b) $\int \frac{x^2 \cot^{-1}(x^3)}{1 + x^6} \, dx$ का मान ज्ञात कीजिए।

c) दो समुच्चयों S और T के लिए दर्शाइए कि:

$S \cup T = (S - T) \cup (S \cap T) \cup (T - S).$

है। वेन आरेख में भी स्थिति दर्शाइए।

5. a) $\mathbf{R}$ पर $f(x) = x^3 - x^2 - 8x + 12$ और
और $g(x) = \begin{cases} \frac{f(x)}{x + 3}, & \text{जब } x \neq -3 \\ \alpha, & \text{जब } x = -3 \end{cases}$

द्वारा परिभाषित दो फलन f और g लीजिए।

i) $\alpha$ का वह मान ज्ञात कीजिए जिसके लिए $g, x = -3$ पर सतत् है।

ii) $f(x) = 0$ के सभी मूल ज्ञात कीजिए।

b) वक्र $y^2(4 - x) = x^3$ और इसकी y-अक्ष के समांतर अनंतस्पर्शी के बीच का क्षेत्रफल ज्ञात कीजिए।

6. a) यदि एक आय फलन $\frac{dR}{dx} = 15 + 2x - x^2$ द्वारा दिया गया है, जहाँ x निवेश है, तो अधिकतम आय ज्ञात कीजिए। यदि प्रारम्भिक आय 0 है, तो आय फलन R भी ज्ञात कीजिए।

b) वक्र $y^2(x + 1) = x^2(3 - x)$ का आरेखण कीजिए और ऐसा करने के लिए प्रयोग किए गये गुणधर्म भी लिखिए।
7. a) चक्रज $x = \alpha(\theta - \sin \theta), y = \alpha(1 - \cos \theta)$ की लम्बाई ज्ञात कीजिए और दर्शाइए कि रेखा $\theta = \frac{2\pi}{3}$ इसे 1 : 3 के अनुपात में विभक्त करती है।

b) वह प्रतिबंध ज्ञात कीजिए कि वक्र $ax^2 + by^2 = 1$ और $a'x^2 + b'y^2 = 1$ एक-दूसरे को लम्बवत् प्रतिच्छेद करते हैं।

8. a) यदि $y = e^{m \sin^{-1} x}$ है, तो दर्शाइए कि $(1 - x^2)y_2 - xy_1 - m^2y = 0$ है। इस प्रकार लाइब्निट्ज के सूत्र का प्रयोग करके (1 - x²)y_n+2 - (2n + 1)xy_n+1 का मान निकालिए।

b) $f(x) = \begin{cases} 2x & , x > 5 \\ x + 5 & , 1 \leq x \leq 5 \\ |x| & , x < 1 \end{cases}$ द्वारा परिभाषित $\mathbf{R}$ फलन $f : \mathbf{R} \to \mathbf{R}$ के जिस भी सबसे बड़े समुच्चय पर सतत् है वह निकालिए।

9. a) समीकरण $x^4 + 15x^3 + 70x^2 + 120x + 64 = 0$ हल कीजिए, जिसके सभी मूल G.P. में हैं।

b) $\lim_{x \to 0} \frac{\tan x - \sin x}{x^3}$ ज्ञात कीजिए।

10. a) यदि $I_{m,n} = \int x^3 (\log x)^n \, dx$ है, तो दर्शाइए कि :

$(m + 1) I_{m,n} = x^{m+1}(\log x)^n - n I_{m,n-1}$ .
है। इस प्रकार $\int x^4 (\log x)^3 \, dx$ ज्ञात कीजिए।

b) $f(x) = 2x^2 - 7x - 10$ द्वारा परिभाषित फलन f के लिए अंतराल [2, 5] पर लैग्रांज माध्यमान प्रमेय सत्यापित कीजिए।

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IGNOU MTE 1 Calculus 2026 Solved Assignment

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IGNOU BSC Assignments Jan - July 2026 - IGNOU University has uploaded its current session Assignment of the BSC Programme for the session year 2026. Students of the BSC Programme can now download Assignment questions from this page. Candidates have to compulsory download those assignments to get a permit of attending the Term End Exam of the IGNOU BSC Programme.

Download a PDF soft copy of IGNOU MTE 1 Calculus BSC Latest Solved Assignment for Session January 2026 - December 2026 in English Language.

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Are you an IGNOU student who wants to download IGNOU Solved Assignment 2024? IGNOU Solved Assignment 2023-24 Session. IGNOU Solved Assignment and In this post, we will provide you with all solved assignments.

If you’ve arrived at this page, you’re looking for a free PDF download of the IGNOU BSC Solved Assignment 2026. BSC is for Bachelor in Science.

IGNOU solved assignments are a set of questions or tasks that students must complete and submit to their respective study centers. The solved assignments are provided by IGNOU Academy and must be completed by the students themselves.

Course Name	Bachelor in Science
Course Code	BSC
Programm	Courses
Language	English

IGNOU MTE 1 Solved Assignment

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Why Choose IGNOU Academy for Your Assignments?

Getting your assignments right is the first step toward a successful degree. At IGNOU Academy, we provide high-quality reference materials designed to simplify your academic journey. Here is why thousands of students trust us:

Latest Curriculum: All content is strictly based on the current IGNOU syllabus.
Perfect Formatting: Understand the ideal structure and layout to score better marks.
Concept Clarity: We break down complex topics into simple, easy-to-grasp explanations.
Exam-Ready: Our materials serve as excellent revision notes for your term-end exams.
Student-Centric Language: Written in clear, simple English/Hindi to ensure every learner understands.
Nationwide Trust: A preferred choice for IGNOU learners across India.

Disclaimer: These materials are intended as reference study guides to help you understand topics and formats. We encourage students to use these insights to prepare and write their own original assignments as per university guidelines.

How to Get Your Solved Assignment PDF

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Step-by-Step: Downloading Official Question Papers

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Click on the "IGNOU Assignment Question Papers" section.
Filter by your Course, Session, and Medium (English/Hindi).
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How to Submit Your IGNOU Assignments

Handwritten is Key: Use clean A4-size sheets and write neatly.
The Front Page: Ensure your first page clearly mentions your Name, Enrollment Number, Course Code, Subject, and Study Center Code.
Offline Submission: Visit your assigned Study Center, submit in person, and always collect your acknowledgment receipt.
Online Submission: If your center allows, scan each subject as a separate PDF. Submit via the official Google Form, Email, or Portal provided by your center. Keep a screenshot of the confirmation.

Tracking Your Submission Status

Want to know if your marks are updated?

Visit the Student Zone on the official IGNOU website.
Navigate to "Assignment Status."
Enter your Enrollment Number and Program Code.
View your submission dates, current status, and any remarks from the evaluator.

A Quick Tip for Success

Dear Students, remember that assignments carry 30% weightage in your final result. They aren't just a formality—they are a game-changer for your overall percentage. Regular study and timely submission are the keys to a high grade.

Success in IGNOU = Smart Study + Well-Prepared Assignments!

Need Help? Contact IGNOU Academy WhatsApp: +91 9199852182 Website: www.ignouacademy.com

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